Andrea Posilicano - Academia.edu (original) (raw)

Papers by Andrea Posilicano

Research paper thumbnail of Relative-Zeta and Casimir Energy for a Semitransparent Hyperplane Selecting Transverse Modes

Advances in Quantum Mechanics, 2017

Research paper thumbnail of Self-adjoint Extensions of Restrictions

We provide a simple recipe for obtaining all self-adjoint extensions, together with their resolve... more We provide a simple recipe for obtaining all self-adjoint extensions, together with their resolvent, of the symmetric operator S obtained by restricting the self-adjoint operator A:(A)⊆→̋ to the dense, closed with respect to the graph norm, subspace ⊂(A). Neither the knowledge of S^* nor of the deficiency spaces of S is required. Typically A is a differential operator and is the kernel of some trace (restriction) operator along a null subset. We parametrise the extensions by the bundle π:()→(), where () denotes the set of orthogonal projections in the Hilbert space ≃(A)/ and π^-1(Π) is the set of self-adjoint operators in the range of Π. The set of self-adjoint operators in , i.e. π^-1(1), parametrises the relatively prime extensions. Any (Π,Θ)∈() determines a boundary condition in the domain of the corresponding extension A_Π,Θ and explicitly appears in the formula for the resolvent (-A_Π,Θ+z)^-1. The connection with both von Neumann's and Boundary Triples theories of self-adjo...

Research paper thumbnail of Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces

The abstract theory of self-adjoint extensions of symmetric operators is used to construct self-a... more The abstract theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic operator on R^n with linear boundary conditions on (a relatively open part of) a compact hypersurface. Our approach allows to obtain Krein-like resolvent formulas where the reference operator coincides with the "free" operator with domain H^2(R^n); this provides an useful tool for the scattering problem from a hypersurface. Concrete examples of this construction are developed in connection with the standard boundary conditions, Dirichlet, Neumann, Robin, δ and δ^'-type, assigned either on a n-1 dimensional compact boundary Γ=∂Ω or on a relatively open part Σ⊂Γ. Schatten-von Neumann estimates for the difference of the powers of resolvents of the free and the perturbed operators are also proven; these give existence and completeness of the wave operators of the associated scattering systems.

Research paper thumbnail of Self-adjoint, globally defined Hamiltonian operators for systems with boundaries

For a general self-adjoint Hamiltonian operator H_0 on the Hilbert space L^2(^d), we determine th... more For a general self-adjoint Hamiltonian operator H_0 on the Hilbert space L^2(^d), we determine the set of all self-adjoint Hamiltonians H on L^2(^d) that dynamically confine the system to an open set Ω⊂^d while reproducing the action of H_0 on an appropriate operator domain. In the case H_0=-Δ +V we construct these Hamiltonians explicitly showing that they can be written in the form H=H_0+ B, where B is a singular boundary potential and H is self-adjoint on its maximal domain. An application to the deformation quantization of one-dimensional systems with boundaries is also presented.

Research paper thumbnail of Direct sums of trace maps and self-adjoint extensions

We give a simple criterion so that a countable infinite direct sum of trace (evaluation) maps is ... more We give a simple criterion so that a countable infinite direct sum of trace (evaluation) maps is a trace map. An application to the theory of self-adjoint extensions of direct sums of symmetric operators is provided; this gives an alternative approach to results recently obtained by Malamud-Neidhardt and Kostenko-Malamud using regularized direct sums of boundary triplets.

Research paper thumbnail of On the spectral theory of Gesztesy-Šeba realizations of 1-D Dirac operators with point interactions on a discrete set

We investigate spectral properties of Gesztesy-Šeba realizations D_X,α and D_X,β of the 1-D Dirac... more We investigate spectral properties of Gesztesy-Šeba realizations D_X,α and D_X,β of the 1-D Dirac differential expression D with point interactions on a discrete set X={x_n}_n=1^∞⊂R. Here α := {α_n}_n=1^∞ and β :={β_n}_n=1^∞⊂R. The Gesztesy-Šeba realizations D_X,α and D_X,β are the relativistic counterparts of the corresponding Schrödinger operators H_X,α and H_X,β with δ- and δ'-interactions, respectively. We define the minimal operator D_X as the direct sum of the minimal Dirac operators on the intervals (x_n-1, x_n). Then using the regularization procedure for direct sum of boundary triplets we construct an appropriate boundary triplet for the maximal operator D_X^* in the case d_*(X):={|x_i-x_j| , i=j} = 0. It turns out that the boundary operators B_X,α and B_X,β parameterizing the realizations D_X,α and D_X,β are Jacobi matrices. These matrices substantially differ from the ones appearing in spectral theory of Schrödinger operators with point interactions. We show that cert...

Research paper thumbnail of Boundary Conditions for Singular Perturbations of Self-Adjoint Operators

Let A:D(A)⊆→̋ be an injective self-adjoint operator and let τ:D(A)→, X a Banach space, be a surje... more Let A:D(A)⊆→̋ be an injective self-adjoint operator and let τ:D(A)→, X a Banach space, be a surjective linear map such that τϕ_< c Aϕ_. Supposing that Range (τ')∩'̋ ={0}, we define a family A^τ_Θ of self-adjoint operators which are extensions of the symmetric operator A_|{τ=0}.. Any ϕ in the operator domain D(A^τ_Θ) is characterized by a sort of boundary conditions on its univocally defined regular component , which belongs to the completion of D(A) w.r.t. the norm Aϕ_. These boundary conditions are written in terms of the map τ, playing the role of a trace (restriction) operator, as τ=Θ Q_ϕ, the extension parameter Θ being a self-adjoint operator from X' to X. The self-adjoint extension is then simply defined by A^τ_Θϕ:=A . The case in which Aϕ=T*ϕ is a convolution operator on LD, T a distribution with compact support, is studied in detail.

Research paper thumbnail of Singular Perturbations of Abstract Wave equations

Given, on the Hilbert space _̋0, the self-adjoint operator B and the skew-adjoint operators C_1 a... more Given, on the Hilbert space _̋0, the self-adjoint operator B and the skew-adjoint operators C_1 and C_2, we consider, on the Hilbert space ≃̋D(B)⊕_̋0, the skew-adjoint operator W=[ C_2& -B^2&C_1] corresponding to the abstract wave equation ϕ̈-(C_1+C_2)ϕ̇=-(B^2+C_1C_2)ϕ. Given then an auxiliary Hilbert space and a linear map τ:D(B^2)→ with a kernel dense in _̋0, we explicitly construct skew-adjoint operators W_Θ on a Hilbert space _̋Θ≃ D(B)⊕_̋0⊕ which coincide with W on ≃⊕ D(B). The extension parameter Θ ranges over the set of positive, bounded and injective self-adjoint operators on . In the case C_1=C_2=0 our construction allows a natural definition of negative (strongly) singular perturbations A_Θ of A:=-B^2 such that the diagram W @>>> W_Θ @AAA @VVV A@>>> A_Θ is commutative.

Research paper thumbnail of A Krein-Like Formula for Singular Perturbations of Self-Adjoint Operators and Applications

Given a self-adjoint operator A:D(A)⊆→ and a continuous linear operator τ:D(A)→ with Range τ'... more Given a self-adjoint operator A:D(A)⊆→ and a continuous linear operator τ:D(A)→ with Range τ'∩' =0, a Banach space, we explicitly construct a family A^τ_Θ of self-adjoint operators such that any A^τ_Θ coincides with the original A on the kernel of τ. Such a family is obtained by giving a Kreĭn-like formula where the role of the deficiency spaces is played by the dual pair (,'); the parameter Θ belongs to the space of symmetric operators from ' to . When = one recovers the "_-2 -construction" of Kiselev and Simon and so, to some extent, our results can be regarded as an extension of it to the infinite rank case. Considering the situation in which =L^2(^n) and τ is the trace (restriction) operator along some null subset, we give various applications to singular perturbations of non necessarily elliptic pseudo-differential operators, thus unifying and extending previously known results.

Research paper thumbnail of On inverses of Krein's Q-functions

Let A_Q be the self-adjoint operator defined by the Q-function Q:z Q_z through the Krein-like res... more Let A_Q be the self-adjoint operator defined by the Q-function Q:z Q_z through the Krein-like resolvent formula (-A_Q+z)^-1= (-A_0+z)^-1+G_zWQ_z^-1VG_z̅^* , z∈ Z_Q , where V and W are bounded operators and Z_Q:={z∈ρ(A_0):Q_z and Q_z̅ have a bounded inverse} . We show that Z_Q=∅ Z_Q=ρ(A_0)∩ρ(A_Q) . We do not suppose that Q is represented in terms of a uniformly strict, operator-valued Nevanlinna function (equivalently, we do not assume that Q is associated to an ordinary boundary triplet), thus our result extends previously known ones. The proof relies on simple algebraic computations stemming from the first resolvent identity.

Research paper thumbnail of Krein's Resolvent Formula for Self-Adjoint Extensions of Symmetric Second Order Elliptic Differential Operators

Given a symmetric, semi-bounded, second order elliptic differential operator on a bounded domain ... more Given a symmetric, semi-bounded, second order elliptic differential operator on a bounded domain with C^1,1 boundary, we provide a Krein-type formula for the resolvent difference between its Friedrichs extension and an arbitrary self-adjoint one.

Research paper thumbnail of Asymptotic flux across hypersurfaces for diffusion processes

We suggest a rigorous definition of the pathwise flux across the boundary of a bounded open set f... more We suggest a rigorous definition of the pathwise flux across the boundary of a bounded open set for transient finite energy diffusion processes. The expectation of such a flux has the property of depending only on the current velocity v, the nonsymmetric (as regards time reversibility) part of the drift. In the case the diffusion has a limiting velocity we then define the asymptotic (as R↑∞) flux across subsets of the sphere of radius R and compute its expectation, again in terms of v.

Research paper thumbnail of Boundary triples and Weyl functions for singular perturbations of self-adjoint operators

Given the symmetric operator A_N obtained by restricting the self-adjoint operator A to N, a line... more Given the symmetric operator A_N obtained by restricting the self-adjoint operator A to N, a linear dense set, closed with respect to the graph norm, we determine a convenient boundary triple for the adjoint A_N^* and the corresponding Weyl function. These objects provide us with the self-adjoint extensions of A_N and their resolvents.

Research paper thumbnail of Scattering into Cones and Flux across Surfaces in Quantum Mechanics: a Pathwise Probabilistic Approach

We show how the scattering-into-cones and flux-across-surfaces theorems in Quantum Mechanics have... more We show how the scattering-into-cones and flux-across-surfaces theorems in Quantum Mechanics have very intuitive pathwise probabilistic versions based on some results by Carlen about large time behaviour of paths of Nelson diffusions. The quantum mechanical results can be then recovered by taking expectations in our pathwise statements.

Research paper thumbnail of On the Common Point Spectrum of Pairs of Self-Adjoint Extensions

Given two different self-adjoint extensions of the same symmetric operator , we analyse the inter... more Given two different self-adjoint extensions of the same symmetric operator , we analyse the intersection of their point spectra. Some simple examples are provided.

Research paper thumbnail of The semiclassical limit on a star-graph with Kirchhoff conditions

arXiv: Mathematical Physics, 2020

We consider the dynamics of a quantum particle of mass mmm on a nnn-edges star-graph with Hamilto... more We consider the dynamics of a quantum particle of mass mmm on a nnn-edges star-graph with Hamiltonian HK=−(2m)−1hbar2DeltaH_K=-(2m)^{-1}\hbar^2 \DeltaHK=−(2m)−1hbar2Delta and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an initial state supported on one of the edges and close to a Gaussian coherent state. We define the limiting classical dynamics through a Liouville operator on the graph, obtained by means of Kre\uin's theory of singular perturbations of self-adjoint operators. For the same class of initial states, we study the semiclassical limit of the wave and scattering operators for the couple (HK,HDoplus)(H_K,H_{D}^{\oplus})(HK,HDoplus), where HDoplusH_{D}^{\oplus}HDoplus is the free Hamiltonian with Dirichlet conditions in the vertex.

Research paper thumbnail of The semi-classical limit with a delta-prime potential

We consider the quantum evolution e t h̄ βψ ξ of a Gaussian coherent state ψ h̄ ξ ∈ L(R) localize... more We consider the quantum evolution e t h̄ βψ ξ of a Gaussian coherent state ψ h̄ ξ ∈ L(R) localized close to the classical state ξ ≡ (q, p) ∈ R, where Hβ denotes a self-adjoint realization of the formal Hamiltonian − h̄ 2 2m d dx2 +βδ ′ 0, with δ ′ 0 the derivative of Dirac’s delta distribution at x = 0 and β a real parameter. We show that in the semi-classical limit such a quantum evolution can be approximated (w.r.t. the L(R)-norm, uniformly for any t ∈ R away from the collision time) by e i h̄ teBφ x , where At = pt 2m , φ x (ξ) := ψ h̄ ξ (x) and LB is a suitable self-adjoint extension of the restriction to C∞ c (M0), M0 := {(q, p)∈R 2 |q 6= 0}, of (−i times) the generator of the free classical dynamics. While the operator LB here utilized is similar to the one appearing in our previous work [2] regarding the semi-classical limit with a delta potential, in the present case the approximation gives a smaller error: it is of order h̄, 0<λ<1/2, whereas it turns out to be of orde...

Research paper thumbnail of Uniqueness in inverse acoustic scattering with unbounded gradient across Lipschitz surfaces

Journal of Differential Equations, Nov 1, 2018

Research paper thumbnail of Scattering from local deformations of a semitransparent plane

Journal of Mathematical Analysis and Applications, 2018

Research paper thumbnail of Limiting absorption principle, generalized eigenfunctions, and scattering matrix for Laplace operators with boundary conditions on hypersurfaces

Journal of Spectral Theory, 2018

Research paper thumbnail of Relative-Zeta and Casimir Energy for a Semitransparent Hyperplane Selecting Transverse Modes

Advances in Quantum Mechanics, 2017

Research paper thumbnail of Self-adjoint Extensions of Restrictions

We provide a simple recipe for obtaining all self-adjoint extensions, together with their resolve... more We provide a simple recipe for obtaining all self-adjoint extensions, together with their resolvent, of the symmetric operator S obtained by restricting the self-adjoint operator A:(A)⊆→̋ to the dense, closed with respect to the graph norm, subspace ⊂(A). Neither the knowledge of S^* nor of the deficiency spaces of S is required. Typically A is a differential operator and is the kernel of some trace (restriction) operator along a null subset. We parametrise the extensions by the bundle π:()→(), where () denotes the set of orthogonal projections in the Hilbert space ≃(A)/ and π^-1(Π) is the set of self-adjoint operators in the range of Π. The set of self-adjoint operators in , i.e. π^-1(1), parametrises the relatively prime extensions. Any (Π,Θ)∈() determines a boundary condition in the domain of the corresponding extension A_Π,Θ and explicitly appears in the formula for the resolvent (-A_Π,Θ+z)^-1. The connection with both von Neumann's and Boundary Triples theories of self-adjo...

Research paper thumbnail of Self-adjoint elliptic operators with boundary conditions on not closed hypersurfaces

The abstract theory of self-adjoint extensions of symmetric operators is used to construct self-a... more The abstract theory of self-adjoint extensions of symmetric operators is used to construct self-adjoint realizations of a second-order elliptic operator on R^n with linear boundary conditions on (a relatively open part of) a compact hypersurface. Our approach allows to obtain Krein-like resolvent formulas where the reference operator coincides with the "free" operator with domain H^2(R^n); this provides an useful tool for the scattering problem from a hypersurface. Concrete examples of this construction are developed in connection with the standard boundary conditions, Dirichlet, Neumann, Robin, δ and δ^'-type, assigned either on a n-1 dimensional compact boundary Γ=∂Ω or on a relatively open part Σ⊂Γ. Schatten-von Neumann estimates for the difference of the powers of resolvents of the free and the perturbed operators are also proven; these give existence and completeness of the wave operators of the associated scattering systems.

Research paper thumbnail of Self-adjoint, globally defined Hamiltonian operators for systems with boundaries

For a general self-adjoint Hamiltonian operator H_0 on the Hilbert space L^2(^d), we determine th... more For a general self-adjoint Hamiltonian operator H_0 on the Hilbert space L^2(^d), we determine the set of all self-adjoint Hamiltonians H on L^2(^d) that dynamically confine the system to an open set Ω⊂^d while reproducing the action of H_0 on an appropriate operator domain. In the case H_0=-Δ +V we construct these Hamiltonians explicitly showing that they can be written in the form H=H_0+ B, where B is a singular boundary potential and H is self-adjoint on its maximal domain. An application to the deformation quantization of one-dimensional systems with boundaries is also presented.

Research paper thumbnail of Direct sums of trace maps and self-adjoint extensions

We give a simple criterion so that a countable infinite direct sum of trace (evaluation) maps is ... more We give a simple criterion so that a countable infinite direct sum of trace (evaluation) maps is a trace map. An application to the theory of self-adjoint extensions of direct sums of symmetric operators is provided; this gives an alternative approach to results recently obtained by Malamud-Neidhardt and Kostenko-Malamud using regularized direct sums of boundary triplets.

Research paper thumbnail of On the spectral theory of Gesztesy-Šeba realizations of 1-D Dirac operators with point interactions on a discrete set

We investigate spectral properties of Gesztesy-Šeba realizations D_X,α and D_X,β of the 1-D Dirac... more We investigate spectral properties of Gesztesy-Šeba realizations D_X,α and D_X,β of the 1-D Dirac differential expression D with point interactions on a discrete set X={x_n}_n=1^∞⊂R. Here α := {α_n}_n=1^∞ and β :={β_n}_n=1^∞⊂R. The Gesztesy-Šeba realizations D_X,α and D_X,β are the relativistic counterparts of the corresponding Schrödinger operators H_X,α and H_X,β with δ- and δ'-interactions, respectively. We define the minimal operator D_X as the direct sum of the minimal Dirac operators on the intervals (x_n-1, x_n). Then using the regularization procedure for direct sum of boundary triplets we construct an appropriate boundary triplet for the maximal operator D_X^* in the case d_*(X):={|x_i-x_j| , i=j} = 0. It turns out that the boundary operators B_X,α and B_X,β parameterizing the realizations D_X,α and D_X,β are Jacobi matrices. These matrices substantially differ from the ones appearing in spectral theory of Schrödinger operators with point interactions. We show that cert...

Research paper thumbnail of Boundary Conditions for Singular Perturbations of Self-Adjoint Operators

Let A:D(A)⊆→̋ be an injective self-adjoint operator and let τ:D(A)→, X a Banach space, be a surje... more Let A:D(A)⊆→̋ be an injective self-adjoint operator and let τ:D(A)→, X a Banach space, be a surjective linear map such that τϕ_< c Aϕ_. Supposing that Range (τ')∩'̋ ={0}, we define a family A^τ_Θ of self-adjoint operators which are extensions of the symmetric operator A_|{τ=0}.. Any ϕ in the operator domain D(A^τ_Θ) is characterized by a sort of boundary conditions on its univocally defined regular component , which belongs to the completion of D(A) w.r.t. the norm Aϕ_. These boundary conditions are written in terms of the map τ, playing the role of a trace (restriction) operator, as τ=Θ Q_ϕ, the extension parameter Θ being a self-adjoint operator from X' to X. The self-adjoint extension is then simply defined by A^τ_Θϕ:=A . The case in which Aϕ=T*ϕ is a convolution operator on LD, T a distribution with compact support, is studied in detail.

Research paper thumbnail of Singular Perturbations of Abstract Wave equations

Given, on the Hilbert space _̋0, the self-adjoint operator B and the skew-adjoint operators C_1 a... more Given, on the Hilbert space _̋0, the self-adjoint operator B and the skew-adjoint operators C_1 and C_2, we consider, on the Hilbert space ≃̋D(B)⊕_̋0, the skew-adjoint operator W=[ C_2& -B^2&C_1] corresponding to the abstract wave equation ϕ̈-(C_1+C_2)ϕ̇=-(B^2+C_1C_2)ϕ. Given then an auxiliary Hilbert space and a linear map τ:D(B^2)→ with a kernel dense in _̋0, we explicitly construct skew-adjoint operators W_Θ on a Hilbert space _̋Θ≃ D(B)⊕_̋0⊕ which coincide with W on ≃⊕ D(B). The extension parameter Θ ranges over the set of positive, bounded and injective self-adjoint operators on . In the case C_1=C_2=0 our construction allows a natural definition of negative (strongly) singular perturbations A_Θ of A:=-B^2 such that the diagram W @>>> W_Θ @AAA @VVV A@>>> A_Θ is commutative.

Research paper thumbnail of A Krein-Like Formula for Singular Perturbations of Self-Adjoint Operators and Applications

Given a self-adjoint operator A:D(A)⊆→ and a continuous linear operator τ:D(A)→ with Range τ'... more Given a self-adjoint operator A:D(A)⊆→ and a continuous linear operator τ:D(A)→ with Range τ'∩' =0, a Banach space, we explicitly construct a family A^τ_Θ of self-adjoint operators such that any A^τ_Θ coincides with the original A on the kernel of τ. Such a family is obtained by giving a Kreĭn-like formula where the role of the deficiency spaces is played by the dual pair (,'); the parameter Θ belongs to the space of symmetric operators from ' to . When = one recovers the "_-2 -construction" of Kiselev and Simon and so, to some extent, our results can be regarded as an extension of it to the infinite rank case. Considering the situation in which =L^2(^n) and τ is the trace (restriction) operator along some null subset, we give various applications to singular perturbations of non necessarily elliptic pseudo-differential operators, thus unifying and extending previously known results.

Research paper thumbnail of On inverses of Krein's Q-functions

Let A_Q be the self-adjoint operator defined by the Q-function Q:z Q_z through the Krein-like res... more Let A_Q be the self-adjoint operator defined by the Q-function Q:z Q_z through the Krein-like resolvent formula (-A_Q+z)^-1= (-A_0+z)^-1+G_zWQ_z^-1VG_z̅^* , z∈ Z_Q , where V and W are bounded operators and Z_Q:={z∈ρ(A_0):Q_z and Q_z̅ have a bounded inverse} . We show that Z_Q=∅ Z_Q=ρ(A_0)∩ρ(A_Q) . We do not suppose that Q is represented in terms of a uniformly strict, operator-valued Nevanlinna function (equivalently, we do not assume that Q is associated to an ordinary boundary triplet), thus our result extends previously known ones. The proof relies on simple algebraic computations stemming from the first resolvent identity.

Research paper thumbnail of Krein's Resolvent Formula for Self-Adjoint Extensions of Symmetric Second Order Elliptic Differential Operators

Given a symmetric, semi-bounded, second order elliptic differential operator on a bounded domain ... more Given a symmetric, semi-bounded, second order elliptic differential operator on a bounded domain with C^1,1 boundary, we provide a Krein-type formula for the resolvent difference between its Friedrichs extension and an arbitrary self-adjoint one.

Research paper thumbnail of Asymptotic flux across hypersurfaces for diffusion processes

We suggest a rigorous definition of the pathwise flux across the boundary of a bounded open set f... more We suggest a rigorous definition of the pathwise flux across the boundary of a bounded open set for transient finite energy diffusion processes. The expectation of such a flux has the property of depending only on the current velocity v, the nonsymmetric (as regards time reversibility) part of the drift. In the case the diffusion has a limiting velocity we then define the asymptotic (as R↑∞) flux across subsets of the sphere of radius R and compute its expectation, again in terms of v.

Research paper thumbnail of Boundary triples and Weyl functions for singular perturbations of self-adjoint operators

Given the symmetric operator A_N obtained by restricting the self-adjoint operator A to N, a line... more Given the symmetric operator A_N obtained by restricting the self-adjoint operator A to N, a linear dense set, closed with respect to the graph norm, we determine a convenient boundary triple for the adjoint A_N^* and the corresponding Weyl function. These objects provide us with the self-adjoint extensions of A_N and their resolvents.

Research paper thumbnail of Scattering into Cones and Flux across Surfaces in Quantum Mechanics: a Pathwise Probabilistic Approach

We show how the scattering-into-cones and flux-across-surfaces theorems in Quantum Mechanics have... more We show how the scattering-into-cones and flux-across-surfaces theorems in Quantum Mechanics have very intuitive pathwise probabilistic versions based on some results by Carlen about large time behaviour of paths of Nelson diffusions. The quantum mechanical results can be then recovered by taking expectations in our pathwise statements.

Research paper thumbnail of On the Common Point Spectrum of Pairs of Self-Adjoint Extensions

Given two different self-adjoint extensions of the same symmetric operator , we analyse the inter... more Given two different self-adjoint extensions of the same symmetric operator , we analyse the intersection of their point spectra. Some simple examples are provided.

Research paper thumbnail of The semiclassical limit on a star-graph with Kirchhoff conditions

arXiv: Mathematical Physics, 2020

We consider the dynamics of a quantum particle of mass mmm on a nnn-edges star-graph with Hamilto... more We consider the dynamics of a quantum particle of mass mmm on a nnn-edges star-graph with Hamiltonian HK=−(2m)−1hbar2DeltaH_K=-(2m)^{-1}\hbar^2 \DeltaHK=−(2m)−1hbar2Delta and Kirchhoff conditions in the vertex. We describe the semiclassical limit of the quantum evolution of an initial state supported on one of the edges and close to a Gaussian coherent state. We define the limiting classical dynamics through a Liouville operator on the graph, obtained by means of Kre\uin's theory of singular perturbations of self-adjoint operators. For the same class of initial states, we study the semiclassical limit of the wave and scattering operators for the couple (HK,HDoplus)(H_K,H_{D}^{\oplus})(HK,HDoplus), where HDoplusH_{D}^{\oplus}HDoplus is the free Hamiltonian with Dirichlet conditions in the vertex.

Research paper thumbnail of The semi-classical limit with a delta-prime potential

We consider the quantum evolution e t h̄ βψ ξ of a Gaussian coherent state ψ h̄ ξ ∈ L(R) localize... more We consider the quantum evolution e t h̄ βψ ξ of a Gaussian coherent state ψ h̄ ξ ∈ L(R) localized close to the classical state ξ ≡ (q, p) ∈ R, where Hβ denotes a self-adjoint realization of the formal Hamiltonian − h̄ 2 2m d dx2 +βδ ′ 0, with δ ′ 0 the derivative of Dirac’s delta distribution at x = 0 and β a real parameter. We show that in the semi-classical limit such a quantum evolution can be approximated (w.r.t. the L(R)-norm, uniformly for any t ∈ R away from the collision time) by e i h̄ teBφ x , where At = pt 2m , φ x (ξ) := ψ h̄ ξ (x) and LB is a suitable self-adjoint extension of the restriction to C∞ c (M0), M0 := {(q, p)∈R 2 |q 6= 0}, of (−i times) the generator of the free classical dynamics. While the operator LB here utilized is similar to the one appearing in our previous work [2] regarding the semi-classical limit with a delta potential, in the present case the approximation gives a smaller error: it is of order h̄, 0<λ<1/2, whereas it turns out to be of orde...

Research paper thumbnail of Uniqueness in inverse acoustic scattering with unbounded gradient across Lipschitz surfaces

Journal of Differential Equations, Nov 1, 2018

Research paper thumbnail of Scattering from local deformations of a semitransparent plane

Journal of Mathematical Analysis and Applications, 2018

Research paper thumbnail of Limiting absorption principle, generalized eigenfunctions, and scattering matrix for Laplace operators with boundary conditions on hypersurfaces

Journal of Spectral Theory, 2018